The series expansion of   near origin is

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Answer (Detailed Solution Below)

Option 3 :

Detailed Solution

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Concept:

Taylor series:

The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number ‘a’ is the power series.

Expression of Taylor series is:

Calculation:

Given:

We have to find the series expansion of  near origin, or a = 0.

Let f(x) = sin x

f(0) = sin (0) = 0,

f'(0) = cos (0) = 1,

f''(0) = -sin(0) = 0,

f'''(0) = -1 .... so on

Putting all the values in Taylor series expansion, we get:

Series expansion of sin x  will be:

Therefore the series expansion of  near origin will be:

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